Question
Seth is trying to pay off his credit card. He plans to pay it off in 24 months. Interest on the card is compounded monthly at a fixed annual rate of 11.6%.
Seth currently has a balance of $18,875 on his credit card. Use the formula for the sum of a finite geometric series to determine Seth’s approximate monthly payment.
Seth currently has a balance of $18,875 on his credit card. Use the formula for the sum of a finite geometric series to determine Seth’s approximate monthly payment.
Answers
GPT-4o mini
To determine Seth's approximate monthly payment to pay off his credit card, we can use the formula for the present value of a series of payments, which is based on the sum of a finite geometric series.
The formula for the present value \( P \) of an ordinary annuity (series of payments) can be represented as:
\[
P = PMT \times \frac{1 - (1 + r)^{-n}}{r}
\]
Where:
- \( P \) = present value (current balance of the credit card)
- \( PMT \) = monthly payment
- \( r \) = monthly interest rate (as a decimal)
- \( n \) = total number of payments
### Step 1: Identify the Variables
1. **Current balance \( P \)**: $18,875
2. **Annual interest rate**: 11.6%
3. **Number of payments \( n \)**: 24 months
4. **Monthly interest rate \( r \)**:
\[
r = \frac{11.6\%}{12} = \frac{0.116}{12} \approx 0.00966667 \text{ (or 0.009666667)}
\]
### Step 2: Rearrange the Formula for \( PMT \)
We want to find \( PMT \). Rearranging the formula gives us:
\[
PMT = P \times \frac{r}{1 - (1 + r)^{-n}}
\]
### Step 3: Plug in the Values
Now we can substitute in the values we have:
\[
PMT = 18875 \times \frac{0.00966667}{1 - (1 + 0.00966667)^{-24}}
\]
Calculating \( (1 + r)^{-n} \):
\[
(1 + 0.00966667)^{-24} \approx (1.00966667)^{-24} \approx 0.785868
\]
Now substituting this back into the equation:
\[
PMT = 18875 \times \frac{0.00966667}{1 - 0.785868}
\]
Calculating the denominator:
\[
1 - 0.785868 \approx 0.214132
\]
Now substitute this into the equation for \( PMT \):
\[
PMT = 18875 \times \frac{0.00966667}{0.214132}
\]
Calculating the fraction:
\[
\frac{0.00966667}{0.214132} \approx 0.04514
\]
Now, calculate \( PMT \):
\[
PMT \approx 18875 \times 0.04514 \approx 851.32
\]
### Final Answer
Seth’s approximate monthly payment to pay off his credit card in 24 months is approximately **$851.32**.
The formula for the present value \( P \) of an ordinary annuity (series of payments) can be represented as:
\[
P = PMT \times \frac{1 - (1 + r)^{-n}}{r}
\]
Where:
- \( P \) = present value (current balance of the credit card)
- \( PMT \) = monthly payment
- \( r \) = monthly interest rate (as a decimal)
- \( n \) = total number of payments
### Step 1: Identify the Variables
1. **Current balance \( P \)**: $18,875
2. **Annual interest rate**: 11.6%
3. **Number of payments \( n \)**: 24 months
4. **Monthly interest rate \( r \)**:
\[
r = \frac{11.6\%}{12} = \frac{0.116}{12} \approx 0.00966667 \text{ (or 0.009666667)}
\]
### Step 2: Rearrange the Formula for \( PMT \)
We want to find \( PMT \). Rearranging the formula gives us:
\[
PMT = P \times \frac{r}{1 - (1 + r)^{-n}}
\]
### Step 3: Plug in the Values
Now we can substitute in the values we have:
\[
PMT = 18875 \times \frac{0.00966667}{1 - (1 + 0.00966667)^{-24}}
\]
Calculating \( (1 + r)^{-n} \):
\[
(1 + 0.00966667)^{-24} \approx (1.00966667)^{-24} \approx 0.785868
\]
Now substituting this back into the equation:
\[
PMT = 18875 \times \frac{0.00966667}{1 - 0.785868}
\]
Calculating the denominator:
\[
1 - 0.785868 \approx 0.214132
\]
Now substitute this into the equation for \( PMT \):
\[
PMT = 18875 \times \frac{0.00966667}{0.214132}
\]
Calculating the fraction:
\[
\frac{0.00966667}{0.214132} \approx 0.04514
\]
Now, calculate \( PMT \):
\[
PMT \approx 18875 \times 0.04514 \approx 851.32
\]
### Final Answer
Seth’s approximate monthly payment to pay off his credit card in 24 months is approximately **$851.32**.